Boundary regularity results for weak solutions of subquadratic elliptic systems [Elektronische Ressource] / vorgelegt von Lisa Beck

Boundary regularity resultsfor weak solutionsof subquadratic elliptic systemsDen Naturwissenschaftlichen Fakult¨atender Friedrich-Alexander-Universitat¨ Erlangen-Nurn¨ bergzurErlangung des Doktorgradesvorgelegt vonLisa Beckaus SchweinfurtAls Dissertation genehmigt von den NaturwissenschaftlichenFakult¨aten der Universit¨at Erlangen-Nurn¨ bergTag der mu¨ndlichen Pru¨fung: 14. Juli 2008Vorsitzender derPromotionskommission: Prof. Dr. E. B¨ anschErstbereichterstatter: Prof. Dr. F. DuzaarZweitberichterstatter: Prof. Dr. A. GasteliZusammenfassung:Die vorliegende Arbeit liefert einen Beitrag zur Regularit¨ atstheorie fu¨r nichtlineare ellipti-sche Systeme partieller Differentialgleichungen zweiter Ordnung. Wir betrachten schwache1,p N 1,p NL¨ osungen u ∈ g + W (Ω,R ) mit vorgeschriebenen Randwerten g ∈ W (Ω,R ) des0inhomogenen elliptischen Systems− diva(·,u,Du) = b(·,u,Du) in Ω1 nfu¨r ein beschr¨anktesC -Gebiet Ω⊂R und Koeffizientena(·,·,·), die den u¨blichen Bedingun-gen bzgl. Stetigkeit, Wachstum und Elliptizit¨ at genugen.¨ Die Inhomogenit¨at b(·,·,·) sei eineCarath´eodory-Funktion, die entweder eine kontrollierbare oder eine naturl¨ iche Wachstums-bedingung erfu¨llt. Unter diesen Voraussetzungen werden vor allem fur¨ den subquadratischenFall 1 < p < 2 h¨ ohere Integrierbarkeits- bzw.
Publié le : mardi 1 janvier 2008
Lecture(s) : 19
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Boundary regularity results
for weak solutions
of subquadratic elliptic systems
Den Naturwissenschaftlichen Fakult¨aten
der Friedrich-Alexander-Universitat¨ Erlangen-Nurn¨ berg
zur
Erlangung des Doktorgrades
vorgelegt von
Lisa Beck
aus SchweinfurtAls Dissertation genehmigt von den Naturwissenschaftlichen
Fakultate¨ n der Universit¨at Erlangen-Nurn¨ berg
Tag der mu¨ndlichen Pru¨fung: 14. Juli 2008
Vorsitzender der
Promotionskommission: Prof. Dr. E. B¨ ansch
Erstbereichterstatter: Prof. Dr. F. Duzaar
Zweitberichterstatter: Prof. Dr. A. Gasteli
Zusammenfassung:
Die vorliegende Arbeit liefert einen Beitrag zur Regularit¨ atstheorie fu¨r nichtlineare ellipti-
sche Systeme partieller Differentialgleichungen zweiter Ordnung. Wir betrachten schwache
1,p N 1,p NL¨ osungen u ∈ g + W (Ω,R ) mit vorgeschriebenen Randwerten g ∈ W (Ω,R ) des
0
inhomogenen elliptischen Systems
− diva(·,u,Du) = b(·,u,Du) in Ω
1 nfu¨r ein beschr¨anktesC -Gebiet Ω⊂R und Koeffizientena(·,·,·), die den u¨blichen Bedingun-
gen bzgl. Stetigkeit, Wachstum und Elliptizit¨ at genugen.¨ Die Inhomogenit¨at b(·,·,·) sei eine
Carath´eodory-Funktion, die entweder eine kontrollierbare oder eine naturl¨ iche Wachstums-
bedingung erfu¨llt. Unter diesen Voraussetzungen werden vor allem fur¨ den subquadratischen
Fall 1 < p < 2 h¨ ohere Integrierbarkeits- bzw. Regularit¨ atsaussagen der folgenden Art (bis
zum Rand von Ω) erzielt:
1,αSind Ω sowie die Randdaten g von der Klasse C , α ∈ (0, 1), und sind die Koeffizienten
H¨ older-stetig mit Exponent α in den ersten beiden Variablen, so geben wir mithilfe der
Methode der A-harmonischen Approximation eine Charakterisierung der regular¨ en Punkte
von Du bis zum Rand. Der Beweis fu¨hrt direkt zur optimalen h¨oheren Regularit¨ at auf der
regulare¨ n Menge (d. h. lokale H¨ older-Stetigkeit von Du zum Exponenten α).
1Fu¨r C -Randwerte g sowie gleichm¨ aßig stetige Koeffizienten zeigen wir Calder´on-Zygmund-
Absch¨atzungen, ein hoheres¨ Integrabilit¨atsresultat, bei dem im Unterschied zu klassischen Re-
sultaten nach Gehring der Gewinn an Integrierbarkeit in quantifizierter Weise bestimmt wird.
H¨ angen die Koeffizienten nicht explizit vonu ab und liegt die Inhomogenit¨ atb(x,u,z)≡b(x)
p/(p−1) q/(p−1) N 1,q N q nNin L , so gilt: b∈L (Ω,R ) und g∈W (Ω,R ) garantieren Du∈L (Ω,R )
np
fu¨r q∈ [p, +δ ] (bzw. q beliebig, falls n = 2).1n−2
In niedrigen Dimensionen n∈ (p,p + 2] beweisen wir außerdem mit der direkten Methode
n−2
und Morrey-Abschatz¨ ungen: u ist lokal H¨ older-stetig zu jedem Exponenten λ∈ (0, 1− )p
außerhalb einer singul¨aren Menge, deren Hausdorffdimension kleiner als n−p ist. Dieses
Resultat gilt sowohl fu¨r nicht-degenerierte als auch fur¨ degenerierte Systeme.
Im letzten Teil der Arbeit beschaftigen¨ wir uns mit Techniken, die eineAbsch¨atzungderHaus-
dorffdimensiondersingular¨ enMenge vonDu in Ω erlauben. Dabei finden alle bisher erzielten
1,αResultate ihre Anwendung. Sind Ω und g von der Klasse C fu¨r ein α∈ (0, 1) und die Ko-
effizienten H¨ older-stetig mit Exponentα in den ersten beiden Variablen, so stellt sich heraus,
dass die Hausdorff-Dimension der singul¨aren Menge vonDu h¨ ochstens min{n−p,n−2α} ist,
1falls n∈ (p,p + 2] erfu¨llt ist. Somit ist insbesondere fu¨r α> fast jeder Randpunkt regular¨
2
(fur¨ eine naturli¨ che Wachstumsbedingung an die Inhomogenit¨ at wird dies nur fur¨ den Fall
p = 2 gezeigt). Ferner gilt dieselbe Aussage fu¨r Koeffizienten der Form a(x,u,z)≡ a(x,z)
unter einer kontrollierbaren Wachstumsbedingung ohne Einschr¨ ankung an die Dimension n.
Der Beweis basiert auf endlichen Differenzen-Operatoren, Interpolationstechniken und ge-
brochenen Sobolev-R¨ aumen. Um dieser Strategie auch am Rand folgen zu k¨ onnen, stellen
wir zwei unterschiedliche Methoden vor: fu¨r kontrollierbares Wachstum gehen wir indirekt
vor und nutzen eine Familie von Vergleichsabbildung, die L¨ osungen eines regularisierten Sys-
tems sind, sowie Calderon-Zygm´ und-Absch¨ atzungen. Fu¨r natu¨rliches Wachstum hingegen
argumentieren wir direkt und verwenden die Tatsache, dass schichtweise gemittelte Koef-
fizienten in normaler Richtung schwach differenzierbar sind.ii
Abstract:
The current thesis makes a contribution to the field of regularity theory of second-order non-
1,p Nlinear elliptic systems. We consider weak solutionsu∈g+W (Ω,R ) of the inhomogeneous0
elliptic system
− diva(·,u,Du) = b(·,u,Du) in Ω
1,p N n 1with prescribed boundary datag∈W (Ω,R ), a bounded domain Ω⊂R of classC and
a vector field a(·,·,·) which satisfies standard continuity, ellipticity and growth conditions.
N nN NThe inhomogeneity b : Ω×R ×R → R is assumed to be a Carath´eodory function
obeying either a controllable or a natural growth condition. Under these assumptions, the
following higher integrability and regularity results (up to the boundary of Ω) are achieved,
mainly for the subquadratic case 1<p< 2:
1,αWe first require that Ω andg are of classC ,α∈ (0, 1), and that the coefficients are H¨ older
continuous with exponent α with respect to the first and second variable. Via the method
ofA-harmonic approximation we give a characterization of regular points for Du up to the
boundary which extends known results to the inhomogeneous case. The proof yields directly
the optimal higher regularity on the regular set (i. e., local H¨ older continuity of Du with
exponent α).
1Provided that the boundary data g is of class C and that the coefficients are uniformly
continuous we then show Calder´on-Zygmund estimates, a higher integrability result that
yields, in contrast to classical higher integrability obtained from the application of Gehring’s
Lemma, a quantified gain in the higher integrability exponent. If the coefficients do not
p/(p−1)depend explicitly on u and if the inhomogeneity b(x,u,z)≡ b(x) belongs to L , then
q/(p−1) N 1,q N q nNthere holds: b ∈ L (Ω,R ) and g ∈ W (Ω,R ) imply Du ∈ L (Ω,R ) for q ∈
np
[p, +δ ] (or q arbitrary if n = 2).1n−2
Moreover, in low dimensions n∈ (p,p + 2], we prove via the direct method and Morrey-type
n−2estimates: u is locally H¨ older continuous with every exponent λ ∈ (0, 1− ) outside a
p
singular set of Hausdorff dimension less than n−p. This result holds true both for non-
degenerate and degenerate systems.
The last part of the thesis is devoted to techniques which allow us to estimate the Hausdorff
dimension of the singular set of Du in Ω. Here, all the result achieved so far are of impor-
1,αtance. Assuming that Ω and g are of class C for some α∈ (0, 1) and that the coefficients
are H¨ older continuous with exponent α with respect to the first and second variable, we
find: The Hausdorff dimension of the singular set of Du does not exceed min{n−p,n− 2α}
1whenever n ∈ (p,p + 2]. In particular, for α > this implies that almost every boundary
2
point is in fact a regular one (for a natural growth condition this is proved only for p = 2).
Furthermore, this conclusion remains valid for coefficients of the form a(x,u,z) ≡ a(x,z)
and inhomogeneities of controllable growth without any restriction on the dimension n. The
proof is based on finite difference operators, interpolation techniques and fractional Sobolev
spaces. To extend this strategy up to the boundary, we present two different methods: for
controllable growth we proceed directly and use a family of comparison maps (which are
solutions of some regularized system) as well as Calder´ on-Zygmund estimates. For natural
growth, however, we argue in a direct way and employ the fact that slicewise mean values
of the coefficients are weakly differentiable in the normal direction.Contents
1 Introduction 1
2 Preliminaries 9
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Morrey and Campanato spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Fractional Sobolev spaces and interpolation . . . . . . . . . . . . . . . . . . . 12
3 Partial regularity for inhomogeneous systems 19
3.1 Structure conditions and results . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 The transformed system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Linear theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 A Caccioppoli inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5 Estimate for the excess quantity . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.6 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 Comparison estimates 61
4.1 A preliminary Caccioppoli-type inequality . . . . . . . . . . . . . . . . . . . . 62
4.2 Inhomogeneous systems with x-dependency . . . . . . . . . . . . . . . . . . . 68
4.3 Homogeneous systems without x-dependency . . . . . . . . . . . . . . . . . . 72
5 Calder´on-Zygmund estimates 81
5.1 Structure conditions and result . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.3 Local integrability estimates in the interior . . . . . . . . . . . . . . . . . . . 88
5.4 Local integrability estimates up to the boundary . . . . . . . . . . . . . . . . 98
5.5 The global higher integrability result . . . . . . . . . . . . . . . . . . . . . . . 101
iiiiv Contents
6 Low dimensions: partial regularity of the solution 107
6.1 Structure conditions and result . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.2 Higher integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3 Decay estimate for the solution . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.4 Proof of Theorem 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7 Existence of regular boundary points I 129
7.1 Structure conditions and results . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.3 A comparison estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.4 A decay estimate and proof of Theorem 7.1 . . . . . . . . . . . . . . . . . . . 139
7.5 Proof of Theorem 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8 Existence of regular boundary points II 149
8.1 Structure conditions and result . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.2 Slicewise mean values and a Caccioppoli inequality . . . . . . . . . . . . . . . 151
8.3 A preliminary estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.4 Higher integrability of finite differences of Du . . . . . . . . . . . . . . . . . . 156
8.5 An estimate for the full derivative . . . . . . . . . . . . . . . . . . . . . . . . 160
8.6 Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
A Additional Lemmas 177
A.1 The function V (ξ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177μ
A.2 Sobolev-Poincar´e inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
A.3 Further technical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
A.4 A global version of Gehring’s Lemma . . . . . . . . . . . . . . . . . . . . . . . 182
List of Symbols 183
References 185Chapter 1
Introduction
Partial differential equations are often motivated by problems from science and serve as sim-
plified models of physical phenomena. In general, we investigate the existence of a solution,
and furthermore, its qualitative properties like regularity and differentiability. An intuitive
example is the solution to the minimal surface equation – such as a soap film realizing the
least surface area amongst all surfaces spanned by a wire. This equation like many other
partial differential equations in science arises from the universal principle that nature favours
states of minimal type or energy. For this reason, partial differential equations have been of
substantial interest for a long time, and they have finally been studied in a systematic way –
independent of practical applications – since the end of the 19th century. One of the crucial
moments was the year 1900 when David Hilbert formulated 23 unsolved mathematical prob-
lems in his famous lecture at the International Congress of Mathematicians in Paris, one of
them being
Are the solutions of regular problems in the calculus of variations necessarily analytic?
In general, this question was answered in the negative, which in turn raised new questions
when trying to obtain regularity results in some weaker sense. One discarded the strategy
to search for classical solutions (i. e., solutions which are sufficiently smooth). Instead, even
in the cases where the previous question is answered in the affirmative, one first looks for
“weak” solutions in suitable Sobolev spaces solving the equation in an integrated form. This
allows to infer the existence of weak solutions via methods from functional analysis like
Galerkin’s method for nonlinear monotone operators. However, in the following we will only
briefly touch existence problems.
Then, in a second step, one is concerned with the regularity properties of these solutions.
Starting from the famous papers of De Giorgi, Nash and Moser [DG57, Nas58, Mos60] the
theory of (scalar-valued) solutions to single equations is by now well-understood. In partic-
ular, it has been shown, under quite general assumptions on the coefficients of the equation,
that solutions are in fact smooth. On the other hand, in the vectorial case counterexamples
of De Giorgi [DG68] and of Giusti and Miranda [GM68b] dating from 1968 have revealed
that solutions to elliptic systems (as well as minima of variational integrals) may develop
singularities even if the coefficients are analytic. Hence, in contrast to equations, we can
only expect partial regularity results for general nonlinear systems, which means that the
solution is regular outside a singular set. Having to abandon full regularity, we are then
interested in estimating the size of the singular set. This will be the main objective of this
thesis, focusing on estimates up to the boundary and the subquadratic setting.
12 Chapter 1. Introduction
The different chapters of this work are mostly self-contained. Thus, we do not provide
an extensive discussion of the historical background of the results in this introduction and
postpone it to the following chapters. For a broader discussion, we refer to Giaquinta’s
monograph [Gia83] and Mingione’s recent survey article [Min06]. Here, we rather concentrate
on giving a rough overview of the results achieved in the current work and how they fit in
the framework of dimension reduction of the singular set. We also give a brief explanation of
some features of the proofs. We will now begin by describing the system under consideration:
n 1Let n,N ∈ N, n ≥ 2, p ∈ (1, 2), and let Ω ⊂ R be a bounded domain of class C . We
1,p Nconsider weak solutions u∈g +W (Ω,R ) of the inhomogeneous elliptic system0
− diva(·,u,Du) = b(·,u,Du) in Ω (1.1)
1,p N N nN nNwith prescribed boundary valuesg∈W (Ω,R ). The vector fielda: Ω×R ×R →R
1is supposed to be of classC with respect to the last variable (possibly apart from the origin)
and to satisfy standard ellipticity and growth conditions
 p−1
2 2 2 |a(x,u,z)| ≤ L μ +|z| ,
p−2 p−2
2 2 2 2 2 22 2ν μ +|z| |λ| ≤ D a(x,u,z)λ·λ ≤ L μ +|z| |λ| ,z
 p−1  2 2 2|a(x,u,z)−a(x,¯ u,¯ z)| ≤ L μ +|z| ω |x−x¯| +|u−u¯|
N nNfor all x,x¯ ∈ Ω, u,u¯ ∈ R and z,λ ∈ R , where 0 < ν ≤ L and μ ∈ [0, 1] are arbitrary
+ Nconstants and ω :R → (0, 1] is a modulus of continuity. The inhomogeneity b : Ω×R ×
nN N
R → R is assumed to be a Carath´eodory function obeying either a controllable or a
natural growth condition, i. e.,
p−1 p
2 2
2 2|b(x,u,z)| ≤ L (1 +|z| ) or |b(x,u,z)| ≤ L (1 +|z| ) .1 2
We want to comment briefly on the weak formulation of the Dirichlet problem (1.1) and a
suitable space for weak solutions depending on which growth condition on the inhomogeneity
is assumed: Here the term weak solution signifies that u solves (1.1) in integrated form,
R R
∞ Ni. e., there holds a(·,u,Du)·Dϕdx = b(·,u,Du)·ϕdx for all ϕ ∈ C (Ω,R ). The0Ω Ω
boundary condition u = g on ∂Ω is to be understood in the sense of traces. In particular,
the existence of second derivatives of u is not required for the weak formulation of (1.1). In
1,p Ngeneral, we shall consider weak solutions in the Sobolev space W (Ω,R ). Then, taking
into account the growth condition on the coefficients and on the inhomogeneity, we note
that the integrals arising in the weak formulation are well-defined and finite. In case of
a natural growth condition, however, we restrict our attention to bounded weak solutions
1,p N ∞ Nu∈ W (Ω,R )∩L (Ω,R ). To justify this restriction, we recall the following example
2 2from [Hil82, Section 2]: Considering the equation 4u = |Du| in B ⊂ R , we observe1/2
that the functions u ≡ 0 and u = log log(1/|x|)− log log 2 are two distinct solutions1 2
1,2in W (B ) both vanishing on the boundary ∂B . A straightforward adaption of this1/2 1/2
1,p Nexample also applies in the subquadratic setting. Hence, taking W (Ω,R ) to be the class
of admissible weak solutions may result in a violation of the “principle of local uniqueness”
which in turn is related to the occurence of irregular weak solutions even in the case of
equations, see also [LU68, Section 1.2]. Apart from boundedness, we will have to assume an
additional smallness condition on the weak solution u. More precisely, we will assume for
the remainder of this introduction that one of the following two conditions holds:3
(1) the inhomogeneity b(·,·,·) obeys a controllable growth condition,
∞ N(2) the inhomogeneity b(·,·,·) obeys a natural growth condition and u∈L (Ω,R ) with
kuk ∞ N ≤M and 2L M <ν.2L (Ω,R )
Keeping in mind these assumptions, we are concerned with the following topics related to
higher integrability and regularity (up to the boundary of Ω):
Partial regularity of Du
We now consider non-degenerate systems (μ> 0) under the assumption of H¨ older continuous
αcoefficients, that is ω(t) = min{1,t } for some α > 0. As mentioned above, passing from
equations to systems (i. e., from N = 1 to N > 1), weak solutions may develop singularities.
Consequently, in a first step, one is interested in proving a partial regularity result, namely
nthat Du is locally H¨ older continuous outside a set ofL -measure zero. For this purpose we
introduce the set of regular points

0 nN
Reg (Ω) := x∈ Ω :Du∈C (U∩ Ω,R ) for a neighbourhood U of xDu
and the set of singular points Sing (Ω) := Ω\ Reg (Ω) of the gradient Du. The proofDu Du
of partial regularity results for nonlinear systems usually relies on a linearization technique
which involves the frozen (linearized) system. Since solutions to linear systems enjoy good
a priori regularity estimates, a comparison principle yields a decay estimate for Du which
is the crucial step in order to control its local behaviour at a given point in Ω. Actually,
there are different proofs of partial regularity, which mainly differ in the implementation
of the linearization described above. By now, these techniques are the indirect approach
via the blow-up technique, the direct approach, and the method ofA-harmonic approxima-
tion. Partial regularity results using these methods were first achieved in the interior (in
the quadratic case) by Morrey, Giusti and Miranda [Mor68, GM68a], Giaquinta, Modica
and Ivert [GM79, Ive79], and Duzaar and Grotowski [DG00], respectively. Furthermore,
Grotowski and Hamburger [Gro00, Ham07] succeeded in extending these techniques up to
the boundary in the (super-)quadratic case and gave a characterization of regular boundary
points (see also [Kro05] for the analogous results concerning almost minimizers of quasicon-
vex variational integrals).
Various subsequent papers were concerned with regularity results for more general nonlinear
systems. We only mention the role of the modulus of continuity ω(·): The assumption of
H¨ older continuity was weakened by Duzaar, Gastel and by Wolf to Dini-continuous coef-
Rr ω(ρ)
ficients requiring merely dρ < ∞ for some r > 0, which still allows to conclude a
0 ρ
partial regularity result for Du, see [DG02, Wol01a]. Assuming merely continuity of the
coefficients, Foss and Mingione [FM08] recently gave a positive answer to the question of low
order partial regularity.
Our first result in this paper is a partial regularity result for inhomogeneous systems with
sublinear growth, stating that Du is in fact not only continuous but H¨ older continuous with
optimal exponent on the set of regular points Reg (Ω), and a characterization of Reg (Ω)Du Du
(see Theorem 3.1 and Theorem 3.2):4 Chapter 1. Introduction
n 1,αTheorem 1.1: Consider p ∈ (1, 2), α ∈ (0, 1), a bounded domain Ω ⊂ R of class C
1,p1,α N Nand g ∈ C (Ω,R ). Let u ∈ g + W (Ω,R ) be a weak solution of (1.1) under the
0
αassumptions stated above with ω(t) = min{1,t }. Then, for y∈ Reg (Ω) there holds: DuDu
is Hol¨der continuous with exponent α in a neighbourhood of y in Ω, and the set of singular
boundary points is contained in Σ ∪ Σ with1 2
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